Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $r = \dfrac{-10}{7(t - 8)} \div \dfrac{-9}{6t(t - 8)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{-10}{7(t - 8)} \times \dfrac{6t(t - 8)}{-9} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ -10 \times 6t(t - 8) } { 7(t - 8) \times -9 } $ $ r = \dfrac{-60t(t - 8)}{-63(t - 8)} $ We can cancel the $t - 8$ so long as $t - 8 \neq 0$ Therefore $t \neq 8$ $r = \dfrac{-60t \cancel{(t - 8})}{-63 \cancel{(t - 8)}} = -\dfrac{60t}{-63} = \dfrac{20t}{21} $